Let $X$ be i.i.d. real random variables such that $E[X_i] = 0$, $\mathrm{Var}[X_i] = \sigma^2$. Moreover it is know that $S_n = \sum_{i=1}^{n}X_i$ is distributed as $x_n X_1 + y_n$ for each $n \geq 1$ and some real $x_n, y_n$. I would like to prove that $X_1$ is distributed as normal distribution $N(0, \sigma^2)$.

1 Answer
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You might want to add as a last step that each random variable $U_n=\frac{S_n}{\sigma\sqrt{n}}$ is distributed as (but not equal to, your post goes too fast on this) the random variable $V_n=\frac{x_n}{\sigma\sqrt{n}}X_1+\frac{y_n}{\sigma\sqrt{n}}$, that $U_n$ converges in distribution to a standard normal random variable, that $V_n$ converges almost surely to $X_1$, and in particular that $V_n$ converges in distribution to $X_1$. Since the limit in distribution is unique, $U_n$ also converges in distribution to $X_1$, hence $X_1$ is indeed standard normal.